CHARACTERISTICS OF RESISTANCE FORCES IN A HYDRAULIC CYLINDER IN DYNAMIC OPERATING MODE

P. Ya. Krauinsh

(Presented by the Joint Seminar of the Departments of Mining Machines and Mechanical-Engineering Technology)

In recent years, hydraulic actuating elements, which have generally recognized advantages, have found wide application as drives in various machines, machine tools, and automatic machines [1, 3, 6, 8, 9, 10]. Many all-Union conferences and meetings on questions of hydraulic drives and hydraulic automation note in their resolutions the urgency of research into dynamic processes in hydraulic drives and of developing methods for accounting for nonlinear processes occurring in hydraulic devices. A number of works [1, 3, 6, 10] note that the static nonlinear characteristics obtained prove to be incorrect when dynamics are studied. It becomes necessary to obtain such characteristics under dynamic conditions.

In the present work an elementary hydraulic drive is considered, taking into account two essential nonlinearities: the nonlinear compliance \(k(p)\) of the connecting flexible control lines as a function of pressure \(p\) [11]; and the nonlinear dependence of the resistance forces \(F_{\mathrm{tr}}\) on the velocity \(V\) of the actuator, which in the dynamic regime differs significantly from the static characteristic.

Fig. 1. Diagram of the experimental setup for studying dynamic processes

The experimental setup of the nonlinear hydraulic system under investigation is shown in Fig. 1. A constant pressure \(p_2\) is supplied to the chamber with area \(F_2\); a flexible hose having compliance \(k(p)\), and a flow generator \(A\), producing a sinusoidal flow \(Q_0 \sin \omega t\), are connected to the chamber with area \(F_1\). A load of mass \(m\), simulating an inertial load, is attached to the rod. The piston–cylinder and rod–cylinder seals were made according to the scheme (Fig. 1—I). With the aid of sensors and other necessary instrumentation, the following quantities were recorded:

\(p_2\) — pressure in the chamber with area \(F_2\);
\(p_1\) — pressure in the chamber with area \(F_1\) and volume \(W_0\);
\(V\) — velocity of the actuator.

The flow generator (a sinusoidal mechanism) made it possible to vary, over wide limits, the amplitude of the input signal \(Q_0\) and the frequency \(\omega\). A typical oscillogram is shown in Fig. 2a. From the oscillogram it is evident that the velocity variation curve consists of three main portions: I — the braking portion, II — the portion of drive stoppage (dwell), III — the acceleration portion. The parameters of the drive under study are known; consequently, from the equation of force balance on the actuator, the resistance force can be determined at any instant of time. The results obtained from processing the oscillogram are shown in Fig. 2b (curve 1), performed both by a numerical method and by a graphical-analytical method [1, 2].

Fig. 2a. Calculated and experimental data for the change in velocity of the actuator

The graphical-analytical method makes it possible to obtain experimental results with sufficient accuracy and with minimal calculations, which is its advantage. Figure 2b also presents the static characteristic of the resistance forces as a function of velocity (curve 2). The resistance forces in the dynamic operating mode differ substantially from their static characteristic. The appearance of a hysteresis loop is evidently caused by the presence, in the gap (Fig. 1—I), of a load-bearing oil film. At the instant of stoppage (portion II — Fig. 2a), sticking of the piston with the rod occurs; as a consequence of this, at the instant of starting there is pure “dry” friction. For the type of seal considered, with increasing velocity it was not possible to obtain significant “falling” portions of the resistance forces reported in [1, 4, 7, 8]. The dependence obtained can be approximated by straight-line segments; moreover, as shown by the results of processing the experimental data, the ratio between the coefficients \(k_1\) and \(k_2\) of the slope of the dynamic characteristic of the resistance forces is approximately equal to the ratio of the areas \(F_1\) and \(F_2\), respectively. Consequently, the reduced characteristic can be determined with sufficient accuracy knowing only the breakaway resistance and the amplitude value of the velocity \(V_a\). Using the recommendations of [3], this characteristic can be written in the following form:

\[ F_{\mathrm{fr}} = \begin{cases} c, & \text{for } \dfrac{dV}{dt} > 0,\quad V>0;\\[6pt] c-k_2\left(V_a - V\right), & \text{for } \dfrac{dV}{dt}<0,\quad 0<V<V_a;\\[6pt] -c, & \text{for } \dfrac{dV}{dt}>0,\quad V<0;\\[6pt] -c+k_2\left(\dfrac{F_1}{F_2}V_a-|V|\right), & \text{for } \dfrac{dV}{dt}<0,\quad -V_a<V<0, \end{cases} \tag{1} \]

where \(c\) is the breakaway friction force.

Fig. 2b. Experimental characteristics of resistance forces

In order to verify the reliability of the experimental results obtained in determining the dynamic characteristic of the resistance forces, a theoretical study of the drive under consideration was carried out. The elementary hydraulic system under consideration, taking into account the nonlinear compliance of the control line \(k(p)\), given in [1], and also obtained by us for the particular system under consideration, is described by the following system of equations:

\[ \left. \begin{aligned} m\frac{dV}{dt}+F_{\mathrm{fr}} &= p_2F_2-p_1F_1;\\[6pt] k(p)W_0\frac{dp_1}{dt}+VF_1 &= Q_0\sin\omega t;\\[6pt] k(p) &= \left(\frac{90}{p_1}+2\right)\cdot 10^{-4}=F(p_1), \end{aligned} \right\} \tag{2} \]

where all quantities are denoted in the text.

Equations (1) and (2) fully describe the dynamic processes in the hydraulic system under consideration under the following assumptions:

a) the amplitude of the input signal is sufficiently small;

b) the frequency of the input signal \(\omega\) varies in the range in which the hyd-

hydraulic shock does not arise; consequently, the inertial heads of the liquid column in the lines may be neglected.

The study of the solutions of the obtained equations may be carried out either by the method of multi-sheet phase portraits (the point-mapping method) [3], or by graph-analytical methods [1, 2], or by modeling on analog machines [1, 10]. The obtained system of equations (1) and (2) was solved by the graph-analytical method [1, 2], taking into account the same parameters, initial conditions, and input signals as in the experimental investigations. The results of the calculations are presented in Fig. 2a (curve 1—experimental velocity data; curve 2—graph-analytical calculation according to equations (1) and (2); curve 3—calculation according to equation (2) and the static characteristic of the resistance forces; Fig. 2b—curve 2).

The data of the analytical calculation according to equations (1) and (2) agree well with the experimental results, which to some extent explains the discrepancies obtained by a number of researchers [3, 4, 5, 7, 8] and others. The data of the calculation according to equation (2) and curve 2 (Fig. 2b) near zero velocity differ substantially from the experimental curve.

This circumstance, evidently, must be taken into account when calculating precision hydraulic-automation systems; moreover, accounting for the resistance forces according to equation (1) does not cause additional difficulties in solving nonlinear equations. It should be noted that the resistance forces (1) are sufficiently stable over a wide range of frequencies of the input signal \(\omega\). At frequencies on the order of \((220 \div 260)\ \mathrm{s}^{-1}\), the resistance force approaches the lines \(d—c—o—a—b\) (Fig. 2b), while zone II (Fig. 2a) disappears; in this case the resistance force may be taken into account by the coefficient of viscous damping [10], equal to

\[ \frac{k_1 + k_2}{2}. \]

REFERENCES

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